Integral question

*To*: mathgroup at smc.vnet.net*Subject*: [mg73675] Integral question*From*: Tulga Ersal <tersal at umich.edu>*Date*: Sat, 24 Feb 2007 02:20:22 -0500 (EST)

Dear Mathematica users, Let's consider the integral Integrate[1/Sqrt[v^2 - c*s^2], s] where v and c are positive reals. When I calculate the integral by hand, I get ArcSin[(Sqrt[c]*s)/v]/Sqrt[c] However, if I evaluate the integral in Mathematica, I get (I*Log[(-2*I)*Sqrt[c]*s + 2*Sqrt[-(c*s^2) + v^2]])/Sqrt[c] which not only does not look like what I have found by hand, but also, for v=4, c=2, s=1, for example, gives 0.255525 + 1.47039 i as opposed to just 0.255525. If you evaluate the integral in Mathematica with the said values (v=4, c=2) Integrate[1/Sqrt[4^2 - 2*s^2], s] you get the answer ArcSin[s/(2*Sqrt[2])]/Sqrt[2] which agrees with what I have found by hand. My question is: How can I get what I found by hand using Mathematica without having to assign values to v and c before evaluating the integral? I tried adding the assumptions v>0, c>0, but it didn't help. I'd appreciate your help. Thanks, Tulga